PHYSICS OF PLASMAS 21, 092305 (2014)
Gyrokinetic modelling of stationary electron and impurity profiles
in tokamaks
A. Skyman,a) D. Tegnered,b) H. Nordman,c) and P. Strandd)
Euratom–VR Association, Department of Earth and Space Sciences, Chalmers University of Technology,
SE-412 96 G€
oteborg, Sweden
(Received 21 February 2014; accepted 21 July 2014; published online 11 September 2014)
Particle transport due to Ion Temperature Gradient (ITG)/Trapped Electron Mode (TEM) turbulence
is investigated using the gyrokinetic code GENE. Both a reduced quasilinear treatment and
nonlinear simulations are performed for typical tokamak parameters corresponding to ITG
dominated turbulence. The gyrokinetic results are compared and contrasted with results from a
computationally efficient fluid model. A selfconsistent treatment is used, where the stationary local
profiles are calculated corresponding to zero particle flux simultaneously for electrons and trace
impurities. The scaling of the stationary profiles with magnetic shear, safety factor, electron-to-ion
temperature ratio, collisionality, toroidal sheared rotation, plasma b, triangularity, and elongation is
investigated. In addition, the effect of different main ion mass on the zero flux condition is
discussed. The electron density gradient can significantly affect the stationary impurity profile
scaling. It is therefore expected that a selfconsistent treatment will yield results more comparable to
experimental results for parameter scans where the stationary background density profile is
sensitive. This is shown to be the case in scans over magnetic shear, collisionality, elongation, and
temperature ratio, for which the simultaneous zero flux electron and impurity profiles are calculated.
A slight asymmetry between hydrogen, deuterium, and tritium with respect to profile peaking is
obtained, in particular, for scans in collisionality and temperature ratio.
[http://dx.doi.org/10.1063/1.4894739]
I. INTRODUCTION
It is well known that the shape of the main ion density
and impurity profiles is crucial for the performance of a
fusion device. Inward peaking of the main ion (electron)
density profile is beneficial for the fusion performance since
it enhances the fusion power production. For impurities on
the other hand, a flat or hollow profile is preferred, since impurity accumulation in the core leads to fuel dilution and
radiation losses which degrades performance.
The particle profiles are determined by a balance between
particle sources and particle fluxes, a subject that historically
has been given much less attention than energy transport and
the associated temperature profiles. Hence, electron density
profiles are often treated as a parameter in theoretical studies
of transport rather than being selfconsistently calculated.
Turbulent transport in the core of tokamaks is expected to
be driven mainly by Ion Temperature Gradient (ITG) and
Trapped Electron (TE) modes. Particle transport driven by
ITG/TE mode turbulence has been investigated in a number of
theoretical studies.1–26 Most work in this area has been focused
on either calculations of stationary electron profiles or on impurity transport using prescribed electron density profiles.
It is well established theoretically that turbulent particle
transport in tokamaks has contributions from both diagonal
(diffusive) and non-diagonal (convective) terms. The nondiagonal transport contributions may give rise to an inward
a)
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c)
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d)
[email protected]. URL: http://www.chalmers.se/rss/.
b)
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pinch which can support an inwardly peaked profile even in
the absence of particle sources in the core. The stationary
peaked profile is then obtained from a balance between diffusion and convection.
It is known that the electron density gradient can significantly affect the stationary impurity profile scaling.24 In the
present work, therefore, the background electron density and
impurity peaking is treated selfconsistently, by simultaneously calculating the local profiles corresponding to zero turbulent particle flux of both electrons and impurities.
Linear and nonlinear gyrokinetic simulations using the
code GENE27 are employed.28,29 The scaling of the stationary profiles with key plasma parameters like magnetic shear,
temperature ratio and temperature gradient, toroidal sheared
rotation, safety factor, and collisionality is investigated for a
deuterium (D) plasma. The isotope scaling of stationary profiles for hydrogen (H), tritium (T), and 50/50 DT plasmas is
also studied. The gyrokinetic results are compared and
contrasted with results from a computationally efficient fluid
model,30 suitable for use in transport codes such as JETTO,
where it is currently implemented.31
The parameters are varied around Cyclone Base Case
(CBC32) parameters, thereby covering a wide range of tokamak scenarios. The CBC is an ITG mode dominated H-mode
discharge and, though set far from marginal stability, is an
interesting case for study, and is widely used as a testing
ground and benchmark for theoretical and numerical studies.
The CBC parameters, with D as main ions in the present
study, are listed in Table I.
The rest of the paper is organised as follows: In Sec. II,
the gyrokinetic and fluid models used in this paper are
21, 092305-1
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Skyman et al.
Phys. Plasmas 21, 092305 (2014)
introduced; in Sec. III, a theoretical background is given,
including considerations regarding analysis and numerics.
The main results are presented in Sec. IV, where scalings of
the stationary profiles for electrons and impurities are discussed; results for background peaking of different main ion
isotopes are presented and discussed in Sec. V; finally, in
Sec. VI, follow the concluding remarks.
II. MODELS
A. GENE
The gyrokinetic simulations in this study were performed using the gyrokinetic turbulence code GENE.28,29
This is an Eulerian-type code of the df variety (i.e., employing a fixed grid in phase space, and considering the perturbed
part of the gyrokinetic distribution function), which allows
for an arbitrary number of species. Both quasi- and nonlinear
simulations including kinetic ions, electrons, and impurities
were performed. All impurities were treated as fully kinetic
species with low concentrations. For studying the effects of
collisions, we considered the GENE implementation of the
Landau–Boltzmann type collision operator.29
For the simulation domain, a flux tube with periodic
boundary conditions in the perpendicular plane was used.
The nonlinear (NL) simulations were performed using a
96 96 32 grid in the normal, bi-normal, and parallel spatial directions, respectively; in the parallel and perpendicular
momentum directions, a 48 12 grid was used. For the linear and quasilinear computations, a typical resolution was
12 24 grid points in the parallel and normal directions,
with 64 12 grid points in momentum space. The nonlinear
simulations were typicallyp
run
up ffito t ¼ 300 R/cs, where R is
ffiffiffiffiffiffiffiffiffiffiffi
the major radius and cs ¼ Te =mi .
effects of finite-Larmor-radius (FLR) have been neglected
for electrons and impurities.
The ion, impurity, and electron perturbations are coupled
through the quasineutrality condition dni =ni þ ZfZ dnZ =nZ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ ft dnet =net þ ð1 ft Þdnet =net , where ft ¼ 2e=ð1 þ eÞ is
the fraction of trapped electrons, and fZ the fraction of impurities, ni, nZ, net, and nef are the densities of the main ions,
impurities, trapped, and free electrons, respectively. The free
electrons are assumed to be Boltzmann distributed. The resulting eigenvalue equation is in the present study reduced to a
set of coupled algebraic equations by assuming a strongly balpffiffiffi
looning eigenfunction (/ ¼ ð1 þ cos hÞ= 3; jhj < p).33 The
operators kk, k?, and the magnetic drift xDj are then replaced
by averages over the strongly ballooning eigenfunction as
1
hkk2 i ¼ ðqRÞ2 ;
3
p2 5 2
2
2
s^ ;
hk? i ¼ kh 1 þ
3 2
2 5
þ s^ ;
hxD i;Z i ¼ xD0 i;Z
3 9
(1)
(2)
(3)
where s^ is the magnetic shear, q is the safety factor, R is the
major radius, and Ð the average over the eigenfunction is
p
defined as h…i ¼ p /ð…Þ/ dh. This analysis is expected
to give results in qualitative agreement with the full eigenvalue solution if magnetic shear is not too small.
The particle and heat fluxes are calculated assuming linear relations between the field quantities combined with a
modified mixing length estimate of the turbulence saturation
level.30 The fluxes are computed numerically by adding the
separate transport contributions from the ITG and TE modes
at a fixed wavenumber, assuming isotropic turbulence with
khqs krqs 0.2–0.3.
B. Fluid model
A brief summary of the fluid model is given here, for a
detailed description, see Refs. 22 and 30 and references
therein. Each species (main ions, electrons, and impurities)
is described by the continuity, parallel momentum, and
energy equations for the perturbations in density, parallel
velocity, and temperature. The parallel velocity perturbation
is zero for the bounce averaged trapped electron fluid, and
TABLE I. Parameters for the CBC.32
r/R
s^
q0
R=Lni;e
R=LTi;e
Ti/Te
Te (keV)
ne (1019 m3)
B0 (T)
R (m)
b (%)a
ei (cs/R)a
a
Denotes derived parameters.
0.18
0.796
1.4
2.22
6.96
1.0
2.85
3.51
3.1
1.65
0.42
0.05
III. STATIONARY PROFILES
The local particle transport for species j can be formally
divided into its diagonal and off-diagonal parts
RCj
R
R
¼ Dj
þ DT j
þ RVp;j :
L nj
LTj
nj
(4)
Here, the first term on the right hand side is the diffusion and
the second and third constitute the off-diagonal pinch. The
first of the pinch terms is the particle transport due to the
temperature gradient (thermo-diffusion) and the second is
the convective velocity, which includes contributions from
curvature, parallel compression and roto-diffusion. In
Eq. (4), R/LXj RrXj/Xj are the local gradient scale
lengths of density and temperature, normalised to the major
radius (R). In general, the transport coefficients dependent on
the gradients, though in the trace impurity limit the transport
is linear in both R=LnZ and R=LTZ . A review of the offdiagonal contributions is given in Ref. 34.
At steady state, the contributions from the different
terms in the particle transport will tend to cancel if the particle sources are small, resulting in zero particle flux. Solving
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Skyman et al.
Eq. (4) for zero particle flux, with Vj ¼ DTj 1=LTj þ Vp;j
yields
R Vj
R ¼
;
(5)
PFj Lnj C¼0
Dj
which is the steady state gradient of zero particle flux for
species j. This measure quantifies the balance between diffusion and advection, and gives a measure of how “peaked”
the local density profile is at steady state. It is therefore
referred to as the “peaking factor” and denoted PFj.
In order to investigate the transport, NL GENE simulations were performed from which PFe for the stationary electron profiles were calculated. The results were compared
with quasilinear (QL) results, also obtained using GENE.
The background peaking factor was found by explicitly seeking the gradient of zero particle flux by calculating the electron flux for several values of the density gradient. A typical
set of simulations is displayed in Fig. 1(a), where the time
evolution of the electron flux for three density gradients near
the gradient of zero particle flux is shown (fluxes are in
gyro-Bohm units, with DGB ¼ cs q2s =R). A second order polynomial p was then fitted to the data closest to the zero flux
gradient and then the PFe was found as the appropriate root
of p. The error for PFe was approximated by finding the corresponding roots of p 6 max[rC], and using the difference
between these roots as a measure of the error. In Fig. 1(b),
the particle flux spectrum for a NL GENE simulation for
CBC near this gradient is shown, together with the corresponding spectrum from the fluid model. The figure illustrates that the total flux is zero due to a balance of inward
and outward transports occurring at different wavenumbers.
This represents a challenge for reduced models using a fixed
wave number, such as the fluid model and QL GENE as used
here. The method for finding PFe from the QL GENE simulations is the same as that described above, but here a
reduced treatment was used, that includes only the dominant
mode, which is an ITG mode for CBC-like parameters.
In the trace impurity limit, i.e., when the fraction of
impurities is sufficiently small, the impurity dynamics do not
affect the turbulence dynamics. Therefore, when finding the
simultaneous peaking factor of the background and impurities, the former can be found first and used in the simulations of the latter without loss of generality. Furthermore, in
the trace impurity limit, the transport coefficients of Eq. (4)
Phys. Plasmas 21, 092305 (2014)
for trace impurities do not depend on the species’ gradients
of density and temperature, meaning that (4) is a linear relation in those gradients. This means that the impurity peaking,
as well as the contribution to PFZ from the thermodiffusion
(PFT) and the convective velocity (PFp), can be found from
simulations with appropriately chosen gradients using the
method outlined in Ref. 35. The peaking factors are calculated for several impurity species, using the reduced QL
GENE model. The impurity peaking factors were also studied using the fluid model, but were seen not to differ significantly from gyrokinetic results and have been left out of the
following figures for clarity. The difference in impurity peaking factors between NL GENE, QL GENE, and the present
fluid model has been covered in previous works.22–24,26
The simulations have been performed in a circular equilibrium with aspect ratio R/a ¼ 3, using kinetic ions, electrons,
and impurities, except when studying the effects of shaping.
Then, the Miller equilibrium model was used instead.36
Impurities were included at trace amounts (nZ/ne ¼ 106), so
as not to affect the turbulent dynamics. The impurity mass was
assumed to be AZ ¼ 2Z, where Z is the charge number. The dynamics were further assumed to be electrostatic (b 0), except
when studying the effect of b on the stationary profiles.
IV. SIMULTANEOUS STATIONARY PROFILES OF
ELECTRONS AND IMPURITIES
First, we examine the dependence of the heat transport
and of PFe on the ion temperature gradient. The result is
shown in Fig. 2, where the ion energy transport from NL
GENE simulations is displayed, together with electron peaking factors from NL GENE, QL GENE, and fluid simulations. The QL GENE and fluid results are obtained for two
different wavenumbers, corresponding to the typical scales
of maximum growth rate (khqs ¼ 0.3) and fluctuation amplitude (khqs ¼ 0.2), respectively. In the TE mode dominated
region (R=LTi ⱗ4:5), the peakedness of the background profile shows a steady increase with increasing ion temperature
gradient, while the ion heat transport is negligible until the
ITG mode is destabilised. Simulations with impurities using
the obtained stationary background gradient were then
performed, choosing R=LTZ ¼ R=LTi . The impurities (Be
(Z ¼ 4), C (Z ¼ 6), Ne (Z ¼ 10), and Ni (Z ¼ 28)) show the
same behaviour, but are considerably less peaked than the
background, at most by a factor of 0.5, as illustrated in Fig.
FIG. 1. Time series and spectra of
electron particle transport for CBC
with D as main ions near to the zero
flux gradient (R=Lne ¼ 2:77); obtained
from NL GENE simulations. A flux
spectrum from the fluid model, renormalised to a peak value of 1, is displayed with the NL GENE spectrum,
and shows the same features.
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Phys. Plasmas 21, 092305 (2014)
FIG. 2. Scaling of background electron
peaking, impurity peaking, ion heat
flux, and eigenvalues with R=LTi .
2(b). Fig. 2(c) illustrates the difference between the TE and
ITG mode scalings: in the TE mode region, both parts of
PFZ are inward, but while the pure convection grows steadily
stronger with increasing R=LTi;Z , the thermopinch contribution decreases in magnitude and eventually changes direction
from inward to outward once R=LTi;Z is sufficiently above the
ITG threshold. Above the threshold, the impurities with
higher Z show less sensitivity to changing R=LTi;Z , which can
be attributed to the dominance for these impurities of the
pure convection part of the peaking factor, which quickly
saturates at PFp,Z 2 above the ITG threshold for all impurities. For the impurities with lower charge numbers, on the
other hand, the peaking factor is dominated by the thermopinch (1/Z), which shows an increasing outward tendency
as the temperature gradient steepens. Though the ion energy
transport shows a stiff increase with the driving gradient
above the ITG threshold, only a moderate reduction is seen
in the peaking factor. The TE mode was seen to pass
smoothly into to the ITG mode, when following zero-flux
density gradient, as shown in Fig. 2(d). The temperature gradient where the mode changes direction therefore corresponds to the real frequency xR ¼ 0. For both ITG and TE
modes, jxR j ! 0 leads to the highest background peaking as
shown in Fig. 2(a) and discussed in Ref. 17. The fluid model
was seen to give the best quantitative agreement with the
nonlinear GENE results for khqs ¼ 0.2, while the tendency
was better reproduced for khqs ¼ 0.3.
It is worth noting that the steady state peaking found in
the simulations is considerably higher than that in the original CBC experiment (R=Lne;i ¼ 2:22). As is known,3,14,17,37
this is due to the neglect of collisions, as they normally are
in the CBC. The collisionality for the CBC parameters is
ei 0.05 cs/R, which is of the same order as the growthrates
and real frequencies observed, and collisions can be expected
to have a notable impact on the transport. When collisions
are added, the background peaking factor is indeed lowered
to a level consistent with the prescribed background gradient
for the CBC, as seen in Fig. 5(a).
The electron peaking factor is reduced with increasing
ion–electron temperature ratio (Ti/Te) for CBC parameters, as
can be seen in Fig. 3. As with the temperature gradient, the
NL GENE results show a strong increase in the ion heat transport, but only a weak scaling of PFe, while the trend is more
pronounced for the QL GENE simulations. The fluid model
shows a good quantitative agreement with the NL GENE
results for khqs ¼ 0.2, while better reproducing the trend for
khqs ¼ 0.3. The discrepancy between the NL and QL GENE
results may be a result of the QL GENE treatment, which
only includes the dominant mode, while the contribution from
the subdominant TE mode is non-negligible for low values of
Ti/Te, resulting in a more moderate scaling for the NL GENE
and fluid models. In Fig. 3(b), the selfconsistently obtained
quasilinear peaking of electrons and impurities is shown.
Impurities with lower charge numbers (Z), as well as the background, show the same dependence on Ti/Te, with a decrease
in the peaking as the ion temperature is increased, and a
weaker tendency for smaller wavenumbers. For the impurities
with higher Z, on the other hand, increased ion temperature
leads to slightly more peaked impurity profiles. In Fig. 3(c), it
is shown that the effect for the impurities is mainly due to an
increase in the relative contribution from the outward thermopinch (1/Z) with increased ion temperature, which affects
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Phys. Plasmas 21, 092305 (2014)
FIG. 3. Scaling of background electron
peaking, impurity peaking, ion heat
flux, and linear eigenvalues with Ti /Te.
the low Z impurities more strongly. To first order, the thermopinch is proportional to the real frequency. As seen in Fig.
3(d) it increases with increasing Ti/Te, which explains its
increasing importance for higher ion temperatures. As was the
case for the scaling with R=LTi , PFe is maximised for xR 0,
as seen when comparing Figs. 3(a) and 3(d).
In Fig. 4(a), the scaling with magnetic shear (^
s ) is studied. The electron peaking shows a strong and near linear
FIG. 4. Scaling of background electron
peaking, impurity peaking, ion heat
flux, and linear eigenvalues with s^.
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Phys. Plasmas 21, 092305 (2014)
FIG. 5. Scaling of background electron
peaking, impurity peaking, ion heat
flux, and linear eigenvalues with ei.
dependence on s^, while the heat transport is only decreased
slightly with increasing magnetic shear. This is similar to the
results reported in Ref. 17 and is due to the shear dependence
of the curvature pinch. This trend is as strong in both the QL
and NL GENE simulations; however, the fluid results show
quantitative and qualitative agreement mostly for khqs ¼ 0.2.
The effect of shear on the linear eigenvalues is not monotonic, with a destabilisation in the low to medium shear
region followed by stabilisation as s^ is increased further. The
selfconsistent results are shown in Fig. 4(b). For the impurities, the change in peaking factors due to magnetic shear follows the trend seen for the electrons, and impurities with
higher Z are more strongly affected. This is seen in Fig. 4(c)
to be due mainly to a stronger inward convective pinch with
increasing shear.
Next we cover the effect of electron–ion collisions on
the peaking factors. Collisionality is known to affect the
background by reducing the peaking factor.3,13,14,17 This is
shown in Fig. 5(a), where a reduction in PFe with increasing
collisionality for all three models is seen, but little or no
effect on the ion heat flux (Fig. 5(d)). The fluid model here
shows a stronger dependence on the collisionality than the
gyrokinetic models, and best reproduces the NL GENE
results at low collisionality using khqs ¼ 0.2 for low collisionality, while khqs ¼ 0.3 agrees better for high collisionality. In Fig. 5(b), the selfconsistent results for a range of
collisionalities are shown. The reduction in peaking factors
with collisionality is also seen for low-Z impurities, while
the high-Z impurities show little or no change in peaking due
to collisions. The effect on the impurities is mainly due to an
increase in the outward thermopinch (1/Z) with increased
collisionality (Fig. 5(c)), due to a change of the real frequency as illustrated in Fig. 5(d).
The influence of sheared toroidal flows on the selfconsistent impurity peaking was also studied. Only purely toroidal
rotation was considered, included through the E B shearing
rate, defined as cE ¼ qr R1 @v@rtor . Hence, we consider flow shear
in the limit where the flow is small, neglecting effects of centrifugal and Coriolis forces. These may, however, be important for heavier impurities.15 The results are shown in Fig.
6(a), where it can be seen that impurities are much more
strongly affected by the rotation than the background, due to
the difference in thermal velocity between main ions and
impurities. For large values of cE, a strong decrease in impurity peaking is seen. The effect is due to the outward rotopinch which becomes important for large values of cE, as
shown in Fig. 6(b). As with the shearing rate, this effect is
more pronounced for high-Z impurities, since the thermopinch
dominates for low Z values. In ASDEX U roto-diffusion has
been found to be a critical ingredient to include in order to
reproduce the Boron profiles seen in experiments.21,25
Next the impact of electro magnetic effects was studied.
As shown in Fig. 6(c), an increase in b leads to a considerably lower background peaking, with PFe decreasing to a
level comparable to PFZ for the higher Z impurities for b of
the order of the experimental value (b ⲏ 0.42%). The self
consistent impurity profiles are less affected, with a
decreased PFZ for low Z impurities, due again to an increase
in the inward thermopinch, as seen in Fig. 6(d). The peaking
factors for the higher charged impurities were less sensitive
to changes in b, though the impurities with the highest values
of Z showed a slight increase in PFZ, owing to an increase in
the pure convection for high values of b. The results are consistent with those discussed in Refs. 20 and 38.
Finally, shaping effects were studied using the Miller
equilibrium model. The quasi-linear electron peaking factor
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Phys. Plasmas 21, 092305 (2014)
FIG. 6. Scaling of background electron
and impurity peaking with E B
shearing rate ((a) and (b)), elongation
((c) and (d)), and b ((e) and (f)).
as well as the self-consistent impurity peaking factors
increase with higher elongation (j) as shown in Fig. 6(e).
For impurities with low charge number, the increase in peaking is mainly due to a larger inward thermopinch while for
high-Z impurities it is caused by an increased pure convection, as seen in Fig. 6(f).
The dependence of the selfconsistent peaking factors on
the safety factor (q0) and triangularity (d) was also studied,
and the scalings were found to be very weak.
V. ISOTOPE EFFECTS ON THE BACKGROUND
PEAKING
The CBC prescribes hydrogen ions as the main ions,
however, for future fusion power plants, a deuterium/tritium
mixture will be used. Due to the difference in mass, it is
known that D and T plasmas will behave differently from
pure H plasmas. Differences in steady state peaking factors
are expected, since both collisions and non-adiabatic electrons can break the gyro-Bohm scaling.39 To get an insight
into the effect of the main ion isotope, the scalings for the
FIG. 7. Eigenvalue spectra (ITG) for CBC parameters (Table I), for H, D,
and T as main ions, with khqs and eigenvalues in species units (cs/R).
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Phys. Plasmas 21, 092305 (2014)
FIG. 8. Scaling of main ion peaking
with different parameters for the CBC
(Table I), for H, D, T, and a 50/50 mixture of D and T as main ions with
khqs ¼ 0.3 in species units.
FIG. 9. Particles fluxes of species D,
T, and e for CBC (Table I) with a 50/
50 mixture of D and T as main ions.
Evaluated at the zero flux gradient for
the pure D case (R=Lne ¼ 2:77) using
NL GENE.
normal CBC were compared with simulations where D was
substituted for H and T.
First, we review the known isotope effects on linear
eigenvalues. Figure 7 displays the ITG eigenvalues in the
collisionless case for H, D, and T in species units. The slight
difference in eigenvalues obtained is due to the
non-adiabatic
pffiffiffiffiffiffiffiffiffiffiffiffi
ffi
electron response into which the mass ratio mi =me enters,
as discussed in Ref. 39.
The QL GENE background peaking versus collisionality
is displayed in Fig. 8(a) for khqsi ¼ 0.3 in species units, corresponding to the peaks in the growthrate spectra. For ei ¼ 0,
a slight difference in PF is observed, with
PFT > PFD > PFH. This is consistent with the asymmetry in
D and T transport reported in Refs. 4 and 40. For larger values of the collisionality, however, the order is reversed.
Next, the effect of the ion mass on the stationary profile
scaling with ion to electron temperature ratio (Ti/Te) is studied. In Fig. 8(b), the peaking factor is seen to decrease
with increasing ion temperature, but in this case the lighter
isotopes are more sensitive, showing a stronger decrease
with Ti/Te. The other parameter scalings discussed in Sec. IV
show only a very weak isotope effect.
The scenario with a 50/50 mixture of D and T was also
studied, and the simultaneous peaking of D and T calculated.
The results were seen to follow the pure D and pure T results
closely, albeit with the T profile approximately 10% more
peaked than the D profile for all values of the collisionality;
see Fig. 8(a). For the scan with Ti/Te, the self-consistent case
gave a larger difference in D and T peaking than the corresponding pure cases, as seen in Fig. 8(b). These results were
corroborated by NL GENE simulations using the standard
CBC parameters, with the background electron density
gradient corresponding to zero flux for the pure D case
(R=Lne ¼ 2:77). The results are shown in Fig. 9. For these
parameters, the electron particle flux remained close to zero,
while the deuterium flux was positive and the tritium flux
negative, indicating a more peaked steady state T profile,
and a less peaked D profile in the mixed scenario. The flow
separation between D and T is also reproduced by the
fluid model, as illustrated in Fig. 10, where the zero flux
for electrons is balanced by a positive D flux and a negative
T flux.
FIG. 10. Particles fluxes of species D, T, and e for CBC (Table I) with a
50/50 mixture of D and T as main ions, evaluated using the fluid model with
khqs ¼ 0.3.
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Skyman et al.
VI. CONCLUSIONS
In the present paper, electron and impurity particles
transport due to ITG/TE mode turbulence was studied using
gyrokinetic simulations and a computationally efficient fluid
model. The gyrokinetic simulations were performed with the
code GENE and both QL and NL GENE simulations were
compared and contrasted with the results of the fluid model.
The focus was on a selfconsistent treatment of particle transport, where the stationary local profiles of electrons and
impurities are calculated simultaneously corresponding to
zero particle flux. The zero flux condition is relevant to the
core region of tokamaks where the particle sources are
absent or small. In addition, the density peaking in plasmas
with varying isotope composition, from pure H, D, and T to
50/50 DT plasmas, was studied. Neoclassical contributions
to the impurity transport, which may be relevant for high-Z
impurities, were neglected. The impurities, with impurity
charge in the region 3 Z 42, were included in low concentrations as trace species. The parameters were taken from
the Cyclone base case, relevant for the core region of tokamaks, and scalings of the stationary profiles with magnetic
shear, safety factor, electron-to-ion temperature ratio, collisionality, sheared toroidal rotation, plasma b, elongation,
and triangularity were investigated.
It was shown that the stationary background density profile was sensitive in scans over magnetic shear, collisionality,
elongation, temperature ratio, and plasma b. The selfconsistent treatment mainly enhanced the trends found in earlier
works on impurity profile peaking assuming a fixed background. For collisionality, however, the increased peaking
found for low collisionality was not accompanied by a
corresponding increase in impurity peaking, making reactor
relevant low collisionality plasma conditions favourable in
this respect. A similar trend was seen see for b, where the
background peaking was seen to be maximised for low b,
while the higher charged impurities were only weakly
affected or showed a lower peaking for lower b.
For safety factor, sheared toroidal rotation and triangularity on the other hand, the effects on the electron profile
were weak and hence a selfconsistent treatment did not add
significant new results to the previous investigations assuming a fixed background. For all considered cases, both the
electron profile and the impurity profile were found to be
inwardly peaked, with stationary density gradient scale
lengths typically in the range R/Ln¼1.0–4.0, i.e., substantially below neoclassical expectations. Furthermore, the electrons were consistently more peaked than the impurities.
In addition, a slight asymmetry between hydrogen,
deuterium, and tritium with respect to profile peaking was
obtained. The effect was more pronounced for high collisionality plasmas and large ion to electron temperature ratios.
The effect of main ion mass on the stationary profiles discussed here is weak, but may result in a D–T fuel separation
in a fusion plasma.
The fluid model was seen to reproduce the gyrokinetic
results both qualitatively and quantitatively, though it was
seen that the reliance one a single wave number for calculating the turbulence was a limitation. The same limitation
Phys. Plasmas 21, 092305 (2014)
applies to the QL GENE model used here, and a more complete QL treatment may give a better agreement.8,17 An
extention of the fluid model to include a spectrum of
wave numbers, as well as effects of shaping and rotation, is
currently under development.
ACKNOWLEDGMENTS
The simulations were performed on resources provided
by the Swedish National Infrastructure for Computing (SNIC)
at PDC Centre for High Performance Computing (PDC-HPC),
on the HELIOS supercomputer system at Computational
Simulation Centre of International Fusion Energy Research
Centre (IFERC-CSC), Aomori, Japan, under the Broader
Approach collaboration between Euratom and Japan,
implemented by Fusion for Energy and JAEA, and on the
supercomputer JUROPA at J€ulich Supercomputing Centre
(JSC). This work was funded by a grant from The Swedish
Research Council (C0338001) and by the European Union’s
Horizon 2020 research and innovation programme under
grant agreement number 633053.
The authors would like to thank F. Jenko, T. G€orler, M.
J. P€uschel, D. Told, and the rest of the GENE team at
IPP–Garching for their valuable support and input.
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